Ambivalent not implies strongly ambivalent

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This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., ambivalent group) need not satisfy the second group property (i.e., strongly ambivalent group)
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It is possible to have an ambivalent group (i.e., a group in which all elements are Real element (?)s) that is not a strongly ambivalent group. In particular, there is at least one element that is not a Strongly real element (?).

As a corollary, a real element of a group need not be a strongly real element.


Example of the quaternion group

Further information: quaternion group

The quaternion group of order eight is an ambivalent group. However, it is not strongly ambivalent, because the only elements of order 2 are \pm 1, and these definitely do not generate the whole group.

Other examples

Other examples include special linear group:SL(2,5).